We suppose through this section that \((M, J, \)  is a smooth toric Fano manifold, \(\overline{\omega }_0 \in 2\pi c_1(M, J)\)  is an initial Kähler form with corresponding barycentered Fano polytope \(({\mathrm {P}}, {{\mathrm {L}}})\) , and we further assume that the Futaki invariant of \((M, J, [\overline{\omega }_0])\)  vanishes. In this case, by [1]}, \((M, J)\)  admits a Kähler-Einstein metric and, by [2]}, [3]}, the Mabuchi energy is coercive relative to the connected component of the identity \(\mathrmsl {Aut}_{0}(M, J)\)  of the group of automorphisms of \((M, J)\) . We first recall what this precisely means. Denote by
\( {\mathcal {H}}(M, \overline{\omega }_0)=\left\lbrace  \varphi \in C^{\infty }(M)\, | \, \overline{\omega }_{\varphi }:= \overline{\omega }_0 + {\mathrmsl d}{\mathrmsl d}^c \varphi >0 \right\rbrace \) 
